Questions

  1. Anscombes quartet is a set of 4 \(x,y\) data sets that were published by Francis Anscombe in a 1973 paper Graphs in statistical analysis. For this first question load the anscombe data that is part of the library(datasets) in R. And assign that data to a new object called data.
data <- anscombe
  1. Summarise the data by calculating the mean, variance, for each column and the correlation between each pair (eg. x1 and y1, x2 and y2, etc) (Hint: use the fBasics() package!)

install.packages(“fBasics”)

library(fBasics)
## Warning: package 'fBasics' was built under R version 3.3.3
## Loading required package: timeDate
## Warning: package 'timeDate' was built under R version 3.3.3
## Loading required package: timeSeries
## Warning: package 'timeSeries' was built under R version 3.3.3
## 
## Rmetrics Package fBasics
## Analysing Markets and calculating Basic Statistics
## Copyright (C) 2005-2014 Rmetrics Association Zurich
## Educational Software for Financial Engineering and Computational Science
## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
## https://www.rmetrics.org --- Mail to: info@rmetrics.org
colAvgs(data)
##       x1       x2       x3       x4       y1       y2       y3       y4 
## 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909 7.500000 7.500909
colVars(data)
##        x1        x2        x3        x4        y1        y2        y3 
## 11.000000 11.000000 11.000000 11.000000  4.127269  4.127629  4.122620 
##        y4 
##  4.123249
paste0("Correlation X1-Y1:",cor(data[,1],data[,5]))
## [1] "Correlation X1-Y1:0.81642051634484"
paste0("Correlation X2-Y2:",cor(data[,2],data[,6]))
## [1] "Correlation X2-Y2:0.816236506000243"
paste0("Correlation X3-Y3:",cor(data[,3],data[,7]))
## [1] "Correlation X3-Y3:0.816286739489598"
paste0("Correlation X4-Y4:",cor(data[,4],data[,8]))
## [1] "Correlation X4-Y4:0.816521436888503"
  1. Create scatter plots for each \(x, y\) pair of data.
plot(data[,1],data[,5], xlab="X1",ylab="Y1",main="Scatterplot X1-Y1")

plot(data[,2],data[,6], xlab="X2",ylab="Y2",main="Scatterplot between x2-y2")

plot(data[,3],data[,7], xlab="X3",ylab="Y3",main="Scatterplot between x3-y3")

plot(data[,4],data[,8], xlab="X4",ylab="Y4",main="Scatterplot between x4-y4")

  1. Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
par(mfrow=c(2,2))
plot(data[,1],data[,5], main="Scatterplot X1-Y1",xlab="X1",ylab="Y1",pch=19) 
plot(data[,2],data[,6], main="Scatterplot X2-Y2",xlab="X2",ylab="Y2",pch=19) 
plot(data[,3],data[,7], main="Scatterplot X3-Y3",xlab="X3",ylab="Y3",pch=19) 
plot(data[,4],data[,8], main="Scatterplot X4-Y4",xlab="X4",ylab="Y4",pch=19)

  1. Now fit a linear model to each data set using the lm() function.
model1<-lm(data[,1]~data[,5])
model2<-lm(data[,2]~data[,6])
model3<-lm(data[,3]~data[,7])
model4<-lm(data[,4]~data[,8])
  1. Now combine the last two tasks. Create a four panel scatter plot matrix that has both the data points and the regression lines. (hint: the model objects will carry over chunks!)
par(mfrow=c(2,2))
plot(data[,1],data[,5], main="Scatterplot X1-Y1",xlab="X1",ylab="Y1",pch=19,abline(model1)) 
plot(data[,2],data[,6], main="Scatterplot X2-Y2",xlab="X2",ylab="Y2",pch=19,abline(model2)) 
plot(data[,3],data[,7], main="Scatterplot X3-Y3",xlab="X3",ylab="Y3",pch=19,abline(model3)) 
plot(data[,4],data[,8], main="Scatterplot X4-Y4",xlab="X4",ylab="Y4",pch=19,abline(model4))

  1. Now compare the model fits for each model object.
summary(model1)

Call: lm(formula = data[, 1] ~ data[, 5])

Residuals: Min 1Q Median 3Q Max -2.6522 -1.5117 -0.2657 1.2341 3.8946

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.9975 2.4344 -0.410 0.69156
data[, 5] 1.3328 0.3142 4.241 0.00217 ** — Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

Residual standard error: 2.019 on 9 degrees of freedom Multiple R-squared: 0.6665, Adjusted R-squared: 0.6295 F-statistic: 17.99 on 1 and 9 DF, p-value: 0.00217

summary(model2)

Call: lm(formula = data[, 2] ~ data[, 6])

Residuals: Min 1Q Median 3Q Max -1.8516 -1.4315 -0.3440 0.8467 4.2017

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.9948 2.4354 -0.408 0.69246
data[, 6] 1.3325 0.3144 4.239 0.00218 ** — Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

Residual standard error: 2.02 on 9 degrees of freedom Multiple R-squared: 0.6662, Adjusted R-squared: 0.6292 F-statistic: 17.97 on 1 and 9 DF, p-value: 0.002179

summary(model3)

Call: lm(formula = data[, 3] ~ data[, 7])

Residuals: Min 1Q Median 3Q Max -2.9869 -1.3733 -0.0266 1.3200 3.2133

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.0003 2.4362 -0.411 0.69097
data[, 7] 1.3334 0.3145 4.239 0.00218 ** — Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

Residual standard error: 2.019 on 9 degrees of freedom Multiple R-squared: 0.6663, Adjusted R-squared: 0.6292 F-statistic: 17.97 on 1 and 9 DF, p-value: 0.002176

summary(model4)

Call: lm(formula = data[, 4] ~ data[, 8])

Residuals: Min 1Q Median 3Q Max -2.7859 -1.4122 -0.1853 1.4551 3.3329

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.0036 2.4349 -0.412 0.68985
data[, 8] 1.3337 0.3143 4.243 0.00216 ** — Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1

Residual standard error: 2.018 on 9 degrees of freedom Multiple R-squared: 0.6667, Adjusted R-squared: 0.6297 F-statistic: 18 on 1 and 9 DF, p-value: 0.002165

  1. In text, summarize the lesson of Anscombe’s Quartet and what it says about the value of data visualization. Data visualization can show some summarized information which can be easily ignored in the summary. From the summary of Models, we can see that their Coefficients, R Squared, Residual Standard errors, and p-values are very close. However, when we look at the plots, we can see that they totally different models. Data Visualization gives us a good overview of the data and a high level picture of the data.